/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem:
 app(app(app(if(),true()),x),y) -> x
 app(app(app(if(),true()),x),y) -> y
 app(app(takeWhile(),p),nil()) -> nil()
 app(app(takeWhile(),p),app(app(cons(),x),xs)) ->
 app(app(app(if(),app(p,x)),app(app(cons(),x),app(app(takeWhile(),p),xs))),nil())
 app(app(dropWhile(),p),nil()) -> nil()
 app(app(dropWhile(),p),app(app(cons(),x),xs)) ->
 app(app(app(if(),app(p,x)),app(app(dropWhile(),p),xs)),app(app(cons(),x),xs))

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   dropWhile ==> dropWhile0/0 dropWhile1/1 dropWhile2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   takeWhile ==> takeWhile0/0 takeWhile1/1 takeWhile2/2
   true ==> true0/0
   if ==> if0/0 if1/1 if2/2 if3/3
   
  uncurry-rules:
   app(if2(x4,x5),x6) -> if3(x4,x5,x6)
   app(if1(x4),x5) -> if2(x4,x5)
   app(if0(),x4) -> if1(x4)
   app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
   app(takeWhile0(),x9) -> takeWhile1(x9)
   app(cons1(x13),x14) -> cons2(x13,x14)
   app(cons0(),x13) -> cons1(x13)
   app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
   app(dropWhile0(),x16) -> dropWhile1(x16)
  eta-rules:
   
  problem:
   if3(true0(),x,y) -> x
   if3(true0(),x,y) -> y
   takeWhile2(p,nil0()) -> nil0()
   takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
   dropWhile2(p,nil0()) -> nil0()
   dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
   app(if2(x4,x5),x6) -> if3(x4,x5,x6)
   app(if1(x4),x5) -> if2(x4,x5)
   app(if0(),x4) -> if1(x4)
   app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
   app(takeWhile0(),x9) -> takeWhile1(x9)
   app(cons1(x13),x14) -> cons2(x13,x14)
   app(cons0(),x13) -> cons1(x13)
   app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
   app(dropWhile0(),x16) -> dropWhile1(x16)
  DP Processor:
   DPs:
    takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
    takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
    takeWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
    dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
    dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
    dropWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),dropWhile2(p,xs),cons2(x,xs))
    app#(if2(x4,x5),x6) -> if{3,#}(x4,x5,x6)
    app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10)
    app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17)
   TRS:
    if3(true0(),x,y) -> x
    if3(true0(),x,y) -> y
    takeWhile2(p,nil0()) -> nil0()
    takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
    dropWhile2(p,nil0()) -> nil0()
    dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
    app(if2(x4,x5),x6) -> if3(x4,x5,x6)
    app(if1(x4),x5) -> if2(x4,x5)
    app(if0(),x4) -> if1(x4)
    app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
    app(takeWhile0(),x9) -> takeWhile1(x9)
    app(cons1(x13),x14) -> cons2(x13,x14)
    app(cons0(),x13) -> cons1(x13)
    app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
    app(dropWhile0(),x16) -> dropWhile1(x16)
   TDG Processor:
    DPs:
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     takeWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     dropWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),dropWhile2(p,xs),cons2(x,xs))
     app#(if2(x4,x5),x6) -> if{3,#}(x4,x5,x6)
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10)
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17)
    TRS:
     if3(true0(),x,y) -> x
     if3(true0(),x,y) -> y
     takeWhile2(p,nil0()) -> nil0()
     takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
     dropWhile2(p,nil0()) -> nil0()
     dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
     app(if2(x4,x5),x6) -> if3(x4,x5,x6)
     app(if1(x4),x5) -> if2(x4,x5)
     app(if0(),x4) -> if1(x4)
     app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
     app(takeWhile0(),x9) -> takeWhile1(x9)
     app(cons1(x13),x14) -> cons2(x13,x14)
     app(cons0(),x13) -> cons1(x13)
     app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
     app(dropWhile0(),x16) -> dropWhile1(x16)
    graph:
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),dropWhile2(p,xs),cons2(x,xs))
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17)
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10)
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(if2(x4,x5),x6) -> if{3,#}(x4,x5,x6)
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),dropWhile2(p,xs),cons2(x,xs))
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17) ->
     dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17)
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10)
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x) ->
     app#(if2(x4,x5),x6) -> if{3,#}(x4,x5,x6)
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> if{3,#}(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs) ->
     takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
    SCC Processor:
     #sccs: 1
     #rules: 6
     #arcs: 18/81
     DPs:
      dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
      dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
      app#(takeWhile1(x9),x10) -> takeWhile{2,#}(x9,x10)
      takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
      takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
      app#(dropWhile1(x16),x17) -> dropWhile{2,#}(x16,x17)
     TRS:
      if3(true0(),x,y) -> x
      if3(true0(),x,y) -> y
      takeWhile2(p,nil0()) -> nil0()
      takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
      dropWhile2(p,nil0()) -> nil0()
      dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
      app(if2(x4,x5),x6) -> if3(x4,x5,x6)
      app(if1(x4),x5) -> if2(x4,x5)
      app(if0(),x4) -> if1(x4)
      app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
      app(takeWhile0(),x9) -> takeWhile1(x9)
      app(cons1(x13),x14) -> cons2(x13,x14)
      app(cons0(),x13) -> cons1(x13)
      app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
      app(dropWhile0(),x16) -> dropWhile1(x16)
     Subterm Criterion Processor:
      simple projection:
       pi(takeWhile{2,#}) = 0
       pi(app#) = 0
       pi(dropWhile{2,#}) = 0
      problem:
       DPs:
        dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
        dropWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
        takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
        takeWhile{2,#}(p,cons2(x,xs)) -> app#(p,x)
       TRS:
        if3(true0(),x,y) -> x
        if3(true0(),x,y) -> y
        takeWhile2(p,nil0()) -> nil0()
        takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
        dropWhile2(p,nil0()) -> nil0()
        dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
        app(if2(x4,x5),x6) -> if3(x4,x5,x6)
        app(if1(x4),x5) -> if2(x4,x5)
        app(if0(),x4) -> if1(x4)
        app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
        app(takeWhile0(),x9) -> takeWhile1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
        app(dropWhile0(),x16) -> dropWhile1(x16)
      SCC Processor:
       #sccs: 2
       #rules: 2
       #arcs: 12/16
       DPs:
        takeWhile{2,#}(p,cons2(x,xs)) -> takeWhile{2,#}(p,xs)
       TRS:
        if3(true0(),x,y) -> x
        if3(true0(),x,y) -> y
        takeWhile2(p,nil0()) -> nil0()
        takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
        dropWhile2(p,nil0()) -> nil0()
        dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
        app(if2(x4,x5),x6) -> if3(x4,x5,x6)
        app(if1(x4),x5) -> if2(x4,x5)
        app(if0(),x4) -> if1(x4)
        app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
        app(takeWhile0(),x9) -> takeWhile1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
        app(dropWhile0(),x16) -> dropWhile1(x16)
       Subterm Criterion Processor:
        simple projection:
         pi(takeWhile{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          if3(true0(),x,y) -> x
          if3(true0(),x,y) -> y
          takeWhile2(p,nil0()) -> nil0()
          takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
          dropWhile2(p,nil0()) -> nil0()
          dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
          app(if2(x4,x5),x6) -> if3(x4,x5,x6)
          app(if1(x4),x5) -> if2(x4,x5)
          app(if0(),x4) -> if1(x4)
          app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
          app(takeWhile0(),x9) -> takeWhile1(x9)
          app(cons1(x13),x14) -> cons2(x13,x14)
          app(cons0(),x13) -> cons1(x13)
          app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
          app(dropWhile0(),x16) -> dropWhile1(x16)
        Qed
       
       DPs:
        dropWhile{2,#}(p,cons2(x,xs)) -> dropWhile{2,#}(p,xs)
       TRS:
        if3(true0(),x,y) -> x
        if3(true0(),x,y) -> y
        takeWhile2(p,nil0()) -> nil0()
        takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
        dropWhile2(p,nil0()) -> nil0()
        dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
        app(if2(x4,x5),x6) -> if3(x4,x5,x6)
        app(if1(x4),x5) -> if2(x4,x5)
        app(if0(),x4) -> if1(x4)
        app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
        app(takeWhile0(),x9) -> takeWhile1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
        app(dropWhile0(),x16) -> dropWhile1(x16)
       Subterm Criterion Processor:
        simple projection:
         pi(dropWhile{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          if3(true0(),x,y) -> x
          if3(true0(),x,y) -> y
          takeWhile2(p,nil0()) -> nil0()
          takeWhile2(p,cons2(x,xs)) -> if3(app(p,x),cons2(x,takeWhile2(p,xs)),nil0())
          dropWhile2(p,nil0()) -> nil0()
          dropWhile2(p,cons2(x,xs)) -> if3(app(p,x),dropWhile2(p,xs),cons2(x,xs))
          app(if2(x4,x5),x6) -> if3(x4,x5,x6)
          app(if1(x4),x5) -> if2(x4,x5)
          app(if0(),x4) -> if1(x4)
          app(takeWhile1(x9),x10) -> takeWhile2(x9,x10)
          app(takeWhile0(),x9) -> takeWhile1(x9)
          app(cons1(x13),x14) -> cons2(x13,x14)
          app(cons0(),x13) -> cons1(x13)
          app(dropWhile1(x16),x17) -> dropWhile2(x16,x17)
          app(dropWhile0(),x16) -> dropWhile1(x16)
        Qed